Problem: Suppose we have a vector-valued function $g(t)$ and a scalar function $f(x, y, z)$. Let $h(t) = f(g(t))$. We know: $\begin{aligned} &g(6) = (3, 4, -5) \\ \\ &g'(6) = (1, 4, 2) \\ \\ &\nabla f(3, 4, -5) = (0, 5, 2) \end{aligned}$ Evaluate $\dfrac{d h}{d t}$ at $t = 6$. $h'(6)=$
Explanation: Formula The multivariable chain rule says that $\dfrac{dh}{dt} = \nabla f(g(t)) \cdot g'(t)$. The $g'(t)$ part is how much a change in $t$ will cause the input to $f$ to move, and the $\nabla f(g(t))$ part is how much $f$ will change in response to this update to its input. [What's the intuition behind the formula?] Applying the formula We want to find $h'(6) = \nabla f(g(6)) \cdot g'(6)$. We know the following. $\begin{aligned} &g(6) = (3, 4, -5) \\ \\ &g'(6) = (1, 4, 2) \\ \\ &\nabla f(3, 4, -5) = (0, 5, 2) \end{aligned}$ Substituting: $h'(6) = (0, 5, 2) \cdot (1, 4, 2) = 24$ Answer Therefore, $h'(6) = 24$.